nach oben

Erschienen in:

2021 | OriginalPaper | Buchkapitel

Curves with Sharp Chabauty-Coleman Bound

verfasst von : Stevan Gajović

Erschienen in: Arithmetic Geometry, Number Theory, and Computation

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We construct curves of each genus g ≥ 2 for which Coleman’s effective Chabauty bound is sharp and Coleman’s theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman’s bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman’s theorem.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves

Nächstes Kapitel Chabauty–Coleman Computations on Rank 1 Picard Curves

[BBM16]

Jennifer S. Balakrishnan, Amnon Besser, and J. Steffen Müller. Quadratic Chabauty: p-adic heights and integral points on hyperelliptic curves. Journal für die reine und angewandte Mathematik, 2016(720):51–79, 01 Nov. 2016.

[BBM17]

Jennifer Balakrishnan, Amnon Besser, and J. Steffen Müller. Computing integral points on hyperelliptic curves using quadratic Chabauty. Mathematics of Computation, 86:1403–1434, 2017.

[BCP97]

Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3–4):235–265, 1997.

[BS09]

Nils Bruin and Michael Stoll. Two-cover descent on hyperelliptic curves. Mathematics of Computation, 78(268):2347–2370, 2009.MathSciNetCrossRef

[BS10]

Nils Bruin and Michael Stoll. The Mordell–Weil sieve: proving non-existence of rational points on curves. LMS Journal of Computation and Mathematics, 13:272–306, 2010.MathSciNetCrossRef

[Cha41]

Claude Chabauty. Sur les points rationnels des courbes algébriques de genre supérieur à l’unité. C. R. Acad. Sci. Paris, 212:882–885, 1941.MathSciNetMATH

[Col85a]

Robert F. Coleman. Effective Chabauty. Duke Math. J., 52(3):765–770, 09 1985.

[Col85b]

Robert F. Coleman. Torsion points on curves and p-adic abelian integrals. Annals of Mathematics, 121(1):111–168, 1985.MathSciNetCrossRef

[Dok04]

Tim Dokchitser. Computing special values of motivic L-functions. Experimental Mathematics, 13(2):137–149, 2004.MathSciNetCrossRef

[Elk]

Noam Elkies. Curves of genus 2 over \(\mathbb {Q}\) whose Jacobians are absolutely simple abelian surfaces with torsion points of high order. http://people.math.harvard.edu/~elkies/g2_tors.html#more.

[Fal83]

Gerd Faltings. Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Inventiones mathematicae, 73(3):349–366, 1983.MathSciNetCrossRef

[GG93]

Daniel M. Gordon and David J. W. Grant. Computing the Mordell-Weil rank of Jacobians of curves of genus two. Transactions of the American Mathematical Society, 337(2):807–824, 1993.

[Gra94]

David Grant. A curve for which Coleman’s effective Chabauty bound is sharp. Proceedings of the American Mathematical Society, 122(1):317–319, 1994.MathSciNetMATH

[HM19]

Yoshinosuke Hirakawa and Hideki Matsumura. A unique pair of triangles. Journal of Number Theory, 194:297–302, 2019.MathSciNetCrossRef

[HS00]

Marc Hindry and Joseph H. Silverman. Diophantine geometry. An Introduction, volume 201 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2000.

[HZ02]

Everett W. Howe and Hui June Zhu. On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field. Journal of Number Theory, 92(1):139–163, 2002.

[JT99]

Kirti Joshi and Pavlos Tzermias. On the Coleman-Chabauty bound. C. R. Acad. Sci. Paris Sér. I Math., 329(6):459–463, 1999.MathSciNetCrossRef

[Kim05]

M. Kim. The motivic fundamental group of \(\mathbb {P}^1\backslash \{0,1,\infty \}\) and the theorem of Siegel. Inventiones mathematicae, 161(3):629–656, 2005.

[Kim09]

Minhyong Kim. The unipotent Albanese map and Selmer varieties for curves. Publ. Res. Inst. Math. Sci., 45(1):89–133, 2009.MathSciNetCrossRef

[KZB13]

Eric Katz and David Zureick-Brown. The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions. Compositio Mathematica, 149(11):1818–1838, 2013.MathSciNetCrossRef

[LT02]

Dino Lorenzini and Thomas J. Tucker. Thue equations and the method of Chabauty-Coleman. Inventiones mathematicae, 148(1):47–77, 2002.MathSciNetCrossRef

[Mor22]

Louis J. Mordell. On the rational solutions of the indeterminate equations of the third and fourth degrees. Proc. Camb. Phil. Soc., 21:179–192, 1922.MATH

[MP12]

William McCallum and Bjorn Poonen. The method of Chabauty and Coleman. Explicit Methods in Number Theory, Panor. Synthèses, 36:99–117, 2012.MathSciNetMATH

[Pau08]

Jennifer Paulhus. Decomposing Jacobians of curves with extra automorphisms. Acta Arith., 132(3):231–244, 2008.MathSciNetCrossRef

[RR96]

Olivier Ramaré and Robert Rumely. Primes in arithmetic progressions. Mathematics of Computation, 65(213):397–425, 1996.MathSciNetCrossRef

[Sch95]

Edward F. Schaefer. 2-descent on the Jacobians of hyperelliptic curves. Journal of Number Theory, 51(2):219–232, 1995.MathSciNetCrossRef

[Sch99]

Victor Scharaschkin. Local-global problems and the Brauer-Manin obstruction. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)–University of Michigan, 1999.

[Sik15]

Samir Siksek. Chabauty and the Mordell-Weil sieve. In Advances on superelliptic curves and their applications, volume 41 of NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., pages 194–224. IOS, Amsterdam, 2015.

[Sto]

Michael Stoll. Genus 2 curves with several points contained in an arithmetic progression. http://www.mathe2.uni-bayreuth.de/stoll/talks/MIT2018-print.pdf.

[Sto01]

Michael Stoll. Implementing 2-descent for Jacobians of hyperelliptic curves. Acta Arith, 98(3):245–277, 2001.MathSciNetCrossRef

[Sto06]

Michael Stoll. Independence of rational points on twists of a given curve. Compositio Mathematica, 142(5), 2006.

[Sto15]

Michael Stoll. Descent and covering collections. In Advances on superelliptic curves and their applications, volume 41 of NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., pages 176–193. IOS, Amsterdam, 2015.

[Tat66]

John Tate. On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Séminaire N. Bourbaki Exp. No. 306, 415–440, 1966.

[Wei29]

André Weil. L’arithmétique sur les courbes algébriques. Acta Math., 52(1):281–315, 1929.MathSciNetCrossRef

Titel: Curves with Sharp Chabauty-Coleman Bound
verfasst von: Stevan Gajović
Verlag: Springer International Publishing
Buch: Arithmetic Geometry, Number Theory, and Computation
Print ISBN: 978-3-030-80913-3

Electronic ISBN: 978-3-030-80914-0

Copyright-Jahr: 2021
DOI: https://doi.org/10.1007/978-3-030-80914-0_15

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"